A Solid Metal Cuboid Of Length 11 Cm Breadth 4 Cm And Height 3 Cm Is Melted To Form Another Solid Consisting Of A Cone And A Cylinder Joint Together As Someone In The Diagram About The Height Of The Corn Is Four Times That Of The Cylinder And Radius Of
Introduction
In this article, we will explore the process of melting a solid metal cuboid and reconstructing it into a cone and a cylinder. The cuboid has a length of 11 cm, a breadth of 4 cm, and a height of 3 cm. We will calculate the volume of the cuboid and then use this information to determine the dimensions of the cone and cylinder that can be formed from it.
Calculating the Volume of the Cuboid
The volume of a cuboid is given by the formula:
V = l × b × h
where V is the volume, l is the length, b is the breadth, and h is the height.
In this case, the length (l) is 11 cm, the breadth (b) is 4 cm, and the height (h) is 3 cm. Therefore, the volume of the cuboid is:
V = 11 × 4 × 3 = 132 cubic cm
The Cone and Cylinder
The melted metal is used to form a cone and a cylinder that are joint together. The height of the cone is four times that of the cylinder. Let's assume the height of the cylinder is x cm. Then, the height of the cone is 4x cm.
The radius of the cone and cylinder is the same. Let's call this radius r cm.
Calculating the Volume of the Cone
The volume of a cone is given by the formula:
V = (1/3) × π × r^2 × h
where V is the volume, π is a constant approximately equal to 3.14, r is the radius, and h is the height.
In this case, the height of the cone is 4x cm, and the radius is r cm. Therefore, the volume of the cone is:
V = (1/3) × π × r^2 × 4x = (4/3) × π × r^2 × x
Calculating the Volume of the Cylinder
The volume of a cylinder is given by the formula:
V = π × r^2 × h
where V is the volume, π is a constant approximately equal to 3.14, r is the radius, and h is the height.
In this case, the height of the cylinder is x cm, and the radius is r cm. Therefore, the volume of the cylinder is:
V = π × r^2 × x
Equating the Volumes
The total volume of the cone and cylinder is equal to the volume of the cuboid. Therefore, we can set up the equation:
(4/3) × π × r^2 × x + π × r^2 × x = 132
Simplifying the equation, we get:
(7/3) × π × r^2 × x = 132
Solving for x
To solve for x, we can divide both sides of the equation by (7/3) × π × r^2:
x = 132 / ((7/3) × π × r^2)
x = 132 / (7/3) × 3.14 × r^2
x = 132 / 22.36 × r^2
x = 5.88 × r^2
Solving for r
To solve for r, we can substitute the value of x into one of the volume equations. Let's use the equation for the cylinder:
V = π × r^2 × x
Substituting the value of x, we get:
132 = π × r^2 × 5.88 × r^2
Simplifying the equation, we get:
132 = 34.56 × π × r^4
Solving for r
To solve for r, we can divide both sides of the equation by 34.56 × π:
r^4 = 132 / (34.56 × π)
r^4 = 132 / 108.24
r^4 = 1.22
r = √[4] 1.22
r = 1.1 cm
Calculating the Height of the Cylinder
Now that we have the value of r, we can calculate the height of the cylinder (x):
x = 5.88 × r^2
x = 5.88 × (1.1)^2
x = 5.88 × 1.21
x = 7.1 cm
Calculating the Height of the Cone
The height of the cone is four times that of the cylinder:
Height of cone = 4 × height of cylinder = 4 × 7.1 = 28.4 cm
Conclusion
Introduction
In our previous article, we explored the process of melting a solid metal cuboid and reconstructing it into a cone and a cylinder. We calculated the volume of the cuboid and then used this information to determine the dimensions of the cone and cylinder that can be formed from it. In this article, we will answer some of the most frequently asked questions related to this topic.
Q: What is the volume of the cuboid?
A: The volume of the cuboid is 132 cubic cm.
Q: What is the radius of the cone and cylinder?
A: The radius of the cone and cylinder is 1.1 cm.
Q: What is the height of the cylinder?
A: The height of the cylinder is 7.1 cm.
Q: What is the height of the cone?
A: The height of the cone is 28.4 cm.
Q: How do we calculate the volume of the cone?
A: The volume of a cone is given by the formula:
V = (1/3) × π × r^2 × h
where V is the volume, π is a constant approximately equal to 3.14, r is the radius, and h is the height.
Q: How do we calculate the volume of the cylinder?
A: The volume of a cylinder is given by the formula:
V = π × r^2 × h
where V is the volume, π is a constant approximately equal to 3.14, r is the radius, and h is the height.
Q: What is the relationship between the height of the cone and the height of the cylinder?
A: The height of the cone is four times that of the cylinder.
Q: How do we solve for x in the equation (7/3) × π × r^2 × x = 132?
A: To solve for x, we can divide both sides of the equation by (7/3) × π × r^2:
x = 132 / ((7/3) × π × r^2)
Q: How do we solve for r in the equation 132 = 34.56 × π × r^4?
A: To solve for r, we can divide both sides of the equation by 34.56 × π:
r^4 = 132 / (34.56 × π)
r^4 = 132 / 108.24
r^4 = 1.22
r = √[4] 1.22
r = 1.1 cm
Q: What is the significance of the radius of the cone and cylinder being the same?
A: The radius of the cone and cylinder being the same means that they have the same base area. This is important because it allows us to calculate the volume of the cone and cylinder using the same formula.
Conclusion
In this article, we have answered some of the most frequently asked questions related to the topic of melting a solid metal cuboid and reconstructing it into a cone and a cylinder. We hope that this article has provided you with a better understanding of the concepts involved and has helped you to clarify any doubts you may have had.